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Abstract

We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.

In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952).
[Crossref]

1957 (1)

1952 (1)

In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952).
[Crossref]

Möllenstedt, G.

Moshinsky, M.

In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952).
[Crossref]

Opt. Express (1)

Phys. Rev. (1)

In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952).
[Crossref]

Figures (4)

Near-field patterns (top) originating from the time evolution of a rectangular wave packet, and their explanation (bottom) with the help of the Cornu spiral. Here we depict the probability density (a) and the phase (b) of the wave function ψ given by Eqs. (4) and (5) in their dependence on the dimensionless time τ and coordinate χ. The white line located at τ = 1/3 indicates both the dominant peak of the probability density corresponding to the focusing of the wave packet and the main plateau of the phase of ψ. The Cornu spirals of (c) and (d) show the complex-valued function F(w) = FC(w) + iFS(w) where the arc length of the curve is parameterized by the argument w. Maxima of |F(w)| correspond to positions on the spiral where the separation from the origin assumes a local maximum. The blue arrows in (c) indicate the first two maxima with n = 0, 1. The corresponding maxima in terms of the space-time variables τ and χ are given by Eqs. (12) and (13). The red arrows indicate maxima of the phase of F(w), which coincide with maxima in the phase of ψ for χ = 0; again we only show the first two corresponding to k = 0, 1. The central arrow in (d) labeled 1 shows the approximate location of the intensity maximum at τ = 1/3 and χ = 0. Arrows 2 and 3 represent arguments of F for τ = 1/3 but with nonzero χ. Since the arguments of the two Fresnel integrals in Eq. (5) defining the wave function ψ are
w=2/τ(1/2±χ), we see that the two contributions are symmetrically placed around χ = 0 and thus ψ has a smaller amplitude but a similar phase compared to ψ at χ = 0.

Probability density |ψ|2 of a freely propagating rectangular wave packet as a function of the space-time variables τ and χ, similar to Fig. 1(a) but on a finer time scale close to the origin. Light and dark colors represent high and low densities, respectively. Superimposed are the parabolas of Eq. (12) and Eq. (13) indicated by blue and red lines, respectively. At the crossings of the parabolas maxima of the intensity pattern occur.

Family of normalized Gaussian widths δ ≡ (κ, τ)/(κ, 0) of the freely-propagating initial rectangular wave packet as a function of the dimensionless time τ ≡ ht/(ma2) and the parametrization κa of the measure. For the optimal parameter κa ≈ 4.5 and τ ≈ 0.3 a global minimum, corresponding to the focused probability peak occurs, in complete agreement with the numerical evaluation of the time-dependent probability density shown in Fig. 1(a) and the analytical considerations of section 2.3. This choice of the parameter κ indicates a maximal shrinkage of the width and therefore represents the best way of capturing focusing.

Influence of sharp edges on the spatio-temporal probability density patterns generated by flat distributions with Gaussian edges. In each frame a different slope is achieved by varying the width Δχ of the Gaussian: Δχ = 1/100 (top left), Δχ = 1/10 (top right), Δχ = 1 (lower left), and Δχ = 10 (lower right). For small values of Δχ, that is for large slopes at the edges we recover the focusing peak and the intricacies of the pattern, while for soft edges, that is for increasing values of Δχ the details near the slit disappear and the focusing effect is mitigated.